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Axiom ax-mulass 7928
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7886. Proofs should normally use mulass 7956 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 7823 . . . 4 class
31, 2wcel 2158 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2158 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2158 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 979 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 7830 . . . . 5 class ·
101, 4, 9co 5888 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 5888 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 5888 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 5888 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1363 . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Colors of variables: wff set class
This axiom is referenced by:  mulass  7956
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